Degenerate homogeneous parabolic equations associated with the infinity-Laplacian

TL;DR

This paper establishes existence, uniqueness, and asymptotic behavior of viscosity solutions for a class of degenerate parabolic equations involving the infinity-Laplacian operator with homogeneity parameter h.

Contribution

It introduces a novel analysis of the degenerate parabolic problem associated with the h-homogeneous infinity-Laplacian, including existence, uniqueness, and asymptotic results.

Findings

Proved existence and uniqueness of viscosity solutions.

Derived asymptotic behavior of solutions in whole space and with zero boundary conditions.

Analyzed the properties of the h-homogeneous infinity-Laplacian operator.

Abstract

We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem $u_t = \Delta_\infty^h u$ where $\Delta_\infty^h$ is the $h$-homogeneous operator associated with the infinity-Laplacian, $\Delta_\infty^h u = |Du|^{h-3} < D^2uDu,Du>$. We also derive the asymptotic behavior of $u$ for the problem posed in the whole space and for the Dirichlet problem with zero boundary conditions.